Primal-Dual Stochastic Hybrid Approximation Algorithm

A new algorithm for solving convex stochastic optimization problems with expectation functions in both the objective and constraints is presented. The algorithm combines a stochastic hybrid procedure, which was originally designed to solve problems with expectation only in the objective, with dual stochastic gradient ascent. More specifically, the algorithm generates primal iterates by minimizing deterministic approximations of the Lagrangian that are updated using noisy subgradients, and dual iterates by applying stochastic gradient ascent to the true Lagrangian. The sequence of primal iterates produced by the algorithm is shown to have a subsequence that converges almost surely to an optimal point under certain conditions. Numerical experience with the new and benchmark algorithms that include a primaldual stochastic approximation algorithm and algorithms based on sample-average approximations is reported. The test problem used originates in power systems operations planning under high penetration of renewable energy, where optimal riskaverse power generation policies are sought. In particular, the problem consists of a two-stage stochastic optimization problem with quadratic objectives and a Conditional Value-At-Risk constraint.